If the volume of a cube is $729 ext{ cm}^3$, then the measure of the main diagonal of the cube will be:

Question

If the volume of a cube is $729 	ext{ cm}^3$, then the measure of the main diagonal of the cube will be:

Examoogle · Accepted Answer

The correct answer is Option C: $9\sqrt{3} 	ext{ cm}$. Since the volume is $729 = 9^3$, the side length is 9. The main diagonal of a cube is always side times $\sqrt{3}$, so the answer is $9\sqrt{3}$.

Step-by-Step Solution & Concepts:
Shortcut Trick: Since the volume is $729 = 9^3$, the side length is 9. The main diagonal of a cube is always side times $\sqrt{3}$, so the answer is $9\sqrt{3}$.

Formula Used: $$V = a^3$$, $$d = a\sqrt{3}$$

Given: The volume of the cube is 729 cm^3.

Steps: Find the side length: The volume $V$ of a cube with side $a$ is given by $V = a^3$. Given $V = 729$, we find the cube root of 729.. Calculate the main diagonal: The formula for the main diagonal $d$ of a cube is $d = a\sqrt{3}$. Substitute $a = 9$ into the formula.

Answer: D is correct because the side length of the cube is 9 cm, which results in a main diagonal of $9\sqrt{3}$ cm.

Exam Tip: Understanding the diagonal relations in 3D shapes is essential for volume and surface area problems in mensuration exams; remember $d = \sqrt{l^2 + w^2 + h^2}$ for cuboids.

Examoogle · Answer

Option A: $7 	ext{ cm}$

Examoogle · Answer

Option B: $7\sqrt{3} 	ext{ cm}$

Examoogle · Answer

Option D: $9	ext{ cm}$

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